TY - GEN A1 - Ferreira, Carlos E. A1 - Martin, Alexander A1 - Weismantel, Robert T1 - Facets for the Multiple Knapsack Problem. N2 - In this paper we consider the multiple knapsack problem which is defined as follows: given a set $N$ of items with weights $f_i$, $i \in N$, a set $M$ of knapsacks with capacities $F_k$, $k \in M$, and a profit function $c_{ik}, i \in N, k \in M$; find an assignment of a subset of the set of items to the set of knapsacks that yields maximum profit (or minimum cost). With every instance of this problem we associate a polyhedron whose vertices are in one to one correspondence to the feasible solutions of the instance. This polytope is the subject of our investigations. In particular, we present several new classes of inequalities and work out necessary and sufficient conditions under which the corresponding inequality defines a facet. Some of these conditions involve only properties of certain knapsack constraints, and hence, apply to the generalized assignment polytope as well. The results presented here serve as the theoretical basis for solving practical problems. The algorithmic side of our study, i.e., separation algorithms, implementation details and computational experience with a branch and cut algorithm are discussed in the companion paper SC 93-07. T3 - ZIB-Report - SC-93-04 Y1 - 1993 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1007 ER - TY - GEN A1 - Martin, Alexander A1 - Weismantel, Robert T1 - Conjunctive Cuts for Integer Programs N2 - This paper deals with a family of conjunctive inequalities. Such inequalities are needed to describe the polyhedron associated with all the integer points that satisfy several knapsack constraints simultaneously. Here we demonstrate the strength and potential of conjunctive inequalities in connection with lifting from a computational point of view. T3 - ZIB-Report - SC-98-18 KW - Polyhedral Combinatorics KW - Integer Programming KW - Separation Algorithm Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3618 ER - TY - GEN A1 - Martin, Alexander A1 - Weismantel, Robert T1 - Contributions to General Mixed Integer Knapsack Problems N2 - This paper deals with a general mixed integer knapsack polyhedron for which we introduce and analyze a new family of inequalities. We discuss the value of this family both from a theoretic and a computational point of view. T3 - ZIB-Report - SC-97-38 Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3075 ER - TY - GEN A1 - Bruns, Winfried A1 - Gubeladze, Joseph A1 - Henk, Martin A1 - Martin, Alexander A1 - Weismantel, Robert T1 - A counterexample to an integer analogue of Caratheodorys theorem N2 - For $n\geq 6$ we provide a counterexample to the conjecture that every integral vector of a $n$-dimensional integral polyhedral pointed cone $C$ can be written as a nonnegative integral combination of at most $n$ elements of the Hilbert basis of $C$. In fact, we show that in general at least $\lfloor 7/6 \cdot n \rfloor$ elements of the Hilbert basis are needed. T3 - ZIB-Report - SC-98-28 KW - Integral pointed cones KW - Hilbert basis KW - integral Carathéodory property Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3718 ER - TY - THES A1 - Weismantel, Robert T1 - Plazieren von Zellen: Theorie und Lösung eines quadratischen 0/1- Optimierungsproblems. N2 - Die vorliegende Arbeit beschäftigt sich mit dem Plazierungsproblem, welches beim Entwurf elektronischer Schaltungen auftritt. Das Plazierungsproblem modellieren wir als ein quadratisches 0/1 Optimierungsproblem unter linearen Nebenbedingungen und untersuchen das Modell komplexitätstheoretisch. Der zweite Aspekt der Arbeit bezieht sich auf die Lösung praktischer Problembeispiele im sogenannten Sea of cells"-Entwurfsstil. Zur Lösung dieser Beispiele wurde ein Prototyp implementiert und mit state of the art"-Plazierungsverfahren verglichen. Schlie\ss lich werden wir uns mit dem Clusteringproblem, das eine Variante des Mehrfachschnitt-Problems darstellt, beschäftigen. Dabei steht einerseits im Vordergrund, wie diese Probleme heuristisch gelöst werden können und wie die Integration des Ansatzes in das Plazierungsprogramm erfolgt. Andererseits soll das Clusteringproblem polyedrisch untersucht werden. T3 - ZIB-Report - TR-92-03 Y1 - 1992 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-4883 ER - TY - GEN A1 - Marchand, Hugues A1 - Martin, Alexander A1 - Weismantel, Robert A1 - Wolsey, Laurence T1 - Cutting Planes in Integer and Mixed Integer Programming N2 - This survey presents cutting planes that are useful or potentially useful in solving mixed integer programs. Valid inequalities for i) general integer programs, ii) problems with local structure such as knapsack constraints, and iii) problems with 0-1 coefficient matrices, such as set packing, are examined in turn. Finally the use of valid inequalities for classes of problems with structure, such as network design, is explored. T3 - ZIB-Report - SC-99-44 KW - Mixed Integer Programming KW - Cutting Planes Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-4339 ER - TY - GEN A1 - Martin, Alexander A1 - Weismantel, Robert T1 - The Intersection of Knapsack Polyhedra and Extensions N2 - This paper introduces a scheme of deriving strong cutting planes for a general integer programming problem. The scheme is related to Chvatal-Gomory cutting planes and important special cases such as odd hole and clique inequalities for the stable set polyhedron or families of inequalities for the knapsack polyhedron. We analyze how relations between covering and incomparability numbers associated with the matrix can be used to bound coefficients in these inequalities. For the intersection of several knapsack polyhedra, incomparabilities between the column vectors of the associated matrix will be shown to transfer into inequalities of the associated polyhedron. Our scheme has been incorporated into the mixed integer programming code SIP. About experimental results will be reported. T3 - ZIB-Report - SC-97-61 Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3300 ER -